Download Geometry Summer Institute 2014 Concept of Congruence and

Geometry Summer Institute 2014
Concept of Congruence and Triangle Proofs
Two geometric figures are congruent if a composition of a finite number of basic rigid
motions maps one to another.
Congruence: the composition of basic rigid motions.
A basic rigid motion maps a geometric figure to a figure that is, intuitively, the same size
and same shape. For this reason, two congruent figures are intuitively the same size and
shape. However, this is not a definition of congruence. The only definition of congruence
between two-dimensional figures is that one can be obtained from the other by a
composition of a finite number of rotations, reflections, and translations.
Congruence of triangles
Because rigid motions preserve distance and angles, two congruent triangles necessarily
have three pairs of equal sides and three pairs of equal angles. The converse is also true:
two triangles with three pairs of equal sides and three pairs of equal angles are congruent.
With triangles, instead of requiring six sets of conditions to guarantee triangle
congruence, a judicious choice of three sets of conditions is sufficient.
Patty Paper Investigation
Side-Side-Side
i.
Draw a large scalene triangle on your patty paper
ii. Copy the three sides separately onto another patty paper, and mark a dot at each
endpoint. Cut the patty paper into three strips with one side on each strip.
iii. Arrange the three segments into a triangle by placing one endpoint on top of
another.
iv.
With a third parry paper trace the formed triangle. Compare the new triangle with
the original one. Are they congruent?
v. Try rearranging the three segments into another triangle. Can you create a triangle
that is not congruent to the original triangle?
vi.
Write a conjecture about two triangles if you know that three sides of one triangle
are congruent to three sides of another triangle.
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Geometry Summer Institute 2014
Angle-Angle-Angle
i.
Construct a large scalene triangle on your patty paper.
ii. Mark a dot in three of the four corners of another patty paper. Copy the three
angles of your original triangle separately onto this second patty paper using each
dot as a vertex of one of the angles. Cut the patty paper into three sections with
one angle on each section. Extend each ray to the edge of the patty paper section.
iii. Arrange the three angles into a triangle by overlapping pairs of rays
iv.
On a third patty paper draw the triangle formed. Compare the new triangle with
the original one. Are they congruent?
v. Try rearranging the three angles into another triangle. Can you create a triangle
that is not congruent to the original triangle?
vi.
Write a conjecture about two triangles if you know that three angles of one
triangle are congruent to three angles of another triangle.
Side-Angle-Side
i.
Construct a large scalene triangle on your patty paper.
ii. Copy any two sides and the angle between them separately onto a second patty
paper. Mark a dot on the vertex of the angle and on each endpoint of each side.
Cut the patty paper into three sections with each part on a different section.
Extend each ray of the copied angle to the edge of its patty paper section.
iii. Arrange the three parts into a triangle
iv.
Place a third patty paper over the three parts, mark a dot at each endpoint, and
draw the triangle formed. Compare the new triangle with the original one. Are
they congruent?
v. Try rearranging the three parts into another triangle. Can you create a triangle that
is not congruent to the original triangle?
vi.
Write a conjecture about two triangles if you know that two sides and the included
angle of one triangle are congruent to two sides and the included angle of another
triangle.
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Geometry Summer Institute 2014
Angle-Side-Angle
i.
Construct a large scalene triangle on your patty paper.
ii. Copy any two angles and the side between them of your original triangle
separately onto a second patty paper as before. Mark a dot on each endpoint and
the vertex of each angle. Cut the patty paper into three sections with each part on
a different section. Extend each ray of each copied angle.
iii. Arrange the three parts into a triangle, placing the side between the two angles
iv.
Place a third patty paper over the three parts, mark a dot at each endpoint and
vertex of each angle, and draw the triangle formed. Compare the new triangle
with the original one. Are they congruent?
v. Try rearranging the three parts into another triangle. Can you create a triangle that
is not congruent to the original triangle?
vi.
Write a conjecture about two triangles if you know that two angles and the
included side of one triangle are congruent to two angles and the included side of
another triangle.
Side-Angle-Angle
i.
Construct a large scalene triangle on your patty paper. Label any two angles and a
side not between the angles.
ii. Copy the two angles and the side not between them of your original triangle
separately onto a second patty paper as before. Mark a dot on each endpoint and
the vertex of each angle. Cut the patty paper into three sections with each part on
a different section. Extend each ray of each copied angle.
iii. Arrange the three parts into a triangle.
iv.
Place a third patty paper over the three parts, mark a dot at each endpoint and
vertex of each angle, and draw the triangle formed. Compare the new triangle
with the original one. Are they congruent?
v. Try rearranging the three parts into another triangle. Can you create a triangle that
is not congruent to the original triangle?
vi.
Write a conjecture about two triangles if you know that two angles and the
included side of one triangle are congruent to two angles and the included side of
another triangle.
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Geometry Summer Institute 2014
Side-Side-Angle
i.
Construct a large acute scalene triangle on your patty paper. Label any two sides
and an angle not between the sides.
ii. Copy the two sides and the angle not between them of your original triangle
separately onto a second patty paper as before. Mark a dot on each endpoint and
the vertex of the angle. Cut the patty paper into three sections with each part on a
different section. Extend each ray of each copied angle.
iii. Arrange the three parts into a triangle, in the same order as the original triangle.
iv.
Place a third patty paper over the three parts, mark a dot at each endpoint and
vertex of each angle, and draw the triangle formed. Compare the new triangle
with the original one. Are they congruent?
v. Try rearranging the three parts into another triangle. Can you create a triangle that
is not congruent to the original triangle?
vi.
Write a conjecture about two triangles if you know that two angles and the
included side of one triangle are congruent to two angles and the included side of
another triangle.
Formal Proofs of the triangle congruence theorems
Proof of the Side-Angle-Side Congruence by basic rigid motions
Suppose we are given triangles ABC and A0B0C0 in the plane so that ∠A and ∠A0 are
equal, and furthermore, |AB| = |A0B0| and |AC| = |A0C0| (see below). We have to explain
why the triangles are congruent.
By our definition of congruence, this means we must exhibit a sequence of basic rigid
motions so that their composition brings (let us say) △ABC to coincide exactly with
△A0B0C0. For ease of comprehension, we will first prove the theorem for the pair of
triangles in the above picture. At the end we will address other possible variations.
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Geometry Summer Institute 2014
We will first move vertex A to A0 by a translation. Let T be the translation along the
vector ⃗⃗⃗⃗⃗⃗⃗
𝐴𝐴0 (from A to A0, shown by the blue vector). We show the image of △ABC by
T in red and use dashed lines to indicate the original positions of △ABC and △A0B0C0.
Next, we will use a rotation to bring the horizontal side of the red triangle (which is the
translated image of AB by T) to A0B0. If the angle between the horizontal red side and
A0B0 is t degrees (in the picture above, t = 90), then a rotation of t degrees around A0 will
map the horizontal ray issuing from A0 to the ray RA0B0. Call this rotation R. Now it is
given that |AB| = |A0B0|, and we know a translation preserves lengths. So the horizontal
side of the red triangle has the same length as A0B0 and therefore R will map the
horizontal side of the red triangle to the side A0B0 of △A0B0C0, as shown.
Two of the vertices of the red triangle already coincide with A0 and B0 of △A0B0C0. We
claim that after a reflection across line LA0B0 the third vertex of the red triangle will be
equal to C0. Indeed, the two marked angles with vertex A0 are equal since basic rigid
motions preserve degrees of angles and, by hypothesis, ∠CAB and ∠C0A0B0 are equal.
Moreover, the left side of the red triangle with A0 as endpoint has the same length as
A0C0 because basic rigid motions preserve length, and by hypothesis |AC| = |A0C0|. Thus
after a reflection across LA0B0, the red triangle coincides with △A0B0C0, as shown:
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Geometry Summer Institute 2014
Thus the desired congruence for the two triangles ABC and A0B0C0 in this particular
picture is the composition of a translation, a rotation, and a reflection.
It remains to address the other possibilities and how they affect the above argument. If A
= A0 to begin with, then the initial translation would be unnecessary. It can also happen
that after the translation T, the image T(AB) (which corresponds to the horizontal side of
the red triangle above) already coincides with A0B0. In that case, the rotation R would be
unnecessary. Finally, if after the rotation the image of C is already on the same side of
LA0B0 as C0, then the image of C and C0 already coincide and the reflection would not be
needed. In any case, theorem SAS is proved.
Now work on your assigned proof of triangle congruence. Be prepared to share with the
whole group.
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